Fitting Spheres in Arbitrary Geometry

I am constructing a MATLAB script to tightly pack an arbitrary given geometry with spheres of a predefined radius. Coordinates of the vertices of the geometry (a polyhedral model) are given; I am thereby developing an algorithm to plot/track all spheres that fit within the boundaries of the model (defined only by the provided vertex data).

I have successfully tested a script that eliminates all spheres that are "crossing" the model boundary; my difficulty, however, lies within teaching MATLAB how to distinguish spheres that are fully within the boundary from spheres that lie entirely exterior to it. What conditions/techniques could I use to (relatively simply) tell MATLAB not to track a sphere if it lies completely outside of the given vertex/boundary data? I aim to construct a grid of spheres that entirely encompasses the geometry and thereafter have the code systematically eliminate spheres that lie upon or outside of the given vertex/boundary data. Only the spheres lying completely within are plotted/tracked!

I seem to be pulling my hair out over this one. Any assistance would be greatly appreciated!